[sdiy] saw vs ramp, audible?

[Donald Tillman]

On Dec 10, 2024, at 1:00 PM, Mattias Rickardsson <mr at analogue.org> wrote:
> 
> Donald Tillman <don at till.com <mailto:don at till.com>> skrev:
>> Mattias Rickardsson <mr at analogue.org <mailto:mr at analogue.org>> wrote:
>>> Donald Tillman <don at till.com <mailto:don at till.com>> skrev:
>>>> Note that a sawtooth ramping down has all the harmonics in phase with the fundamental.
>>> 
>>> 
>>> Yes.
>>> 
>>>> And a sawtooth ramping up has its harmonics alternating in phase (+1, -1/2, +1/3, -1/4,...) from the fundamental.
>>> 
>>> 
>>> But are you counting from the midpoint now? Surely they must still be in phase if you just flip the sawtooth backwards around its discontinuity?
>> 
>> (I had to think about this...)
>> 
>> Good point!  When you play it backwards the even harmonics reverse polarity but the odd harmonics don't.  So yeah, it's consistent that way.
> 
> 
> Not sure how you mean now, but... :-)

Yeah, I goofed... I guess I meant to say that when you flip it backwards the sine components reverse polarity but the cosines don't.

(Sheeshe, you'd think I'd get that right after all those records I played backwards back in the day.  So yeah, if you play a sawtooth wave backwards you hear a secret message.)

And if you were mixing it with a square wave, the square wave would also reverse polarity going backwards.

---

So, the traditional slope-up sawtooth that steps down at t=0 is:

    saw = -sin(t) - (1/2)sin(2t) - (1/3)sin(3t) - (1/4)sin(4t) -...

Which is problematic when you want to mix it with another wave because the fundamental is out of phase.  

You can shift it over so that it steps down halfway through, but then you have alternating harmonic polarities:

    saw = sin(t) - (1/2)sin(2t) + (1/3)sin(3t) - (1/4)sin(4t) +...

My choice, because I care about phases, is a slope-down sawtooth that steps up at t=0, and that's completely in phase:

    saw = sin(t) + (1/2)sin(2t) + (1/3)sin(3t) + (1/4)sin(4t) +...

That is, when you start with a description of the spectrum you want, and figure out the waveform from that, the result is a slope-down sawtooth.

  -- Don
--
Donald Tillman, Palo Alto, California
https://till.com
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